% Mizar ND problem: t12_domain_1,domain_1,51,66 
fof(dh_c1_2__domain_1,definition,
    ( ( ~ v1_xboole_0(c1_2__domain_1)
     => ! [A] : 
          ( ~ v1_xboole_0(A)
         => ! [B] : 
              ( m1_subset_1(B,k2_zfmisc_1(c1_2__domain_1,A))
             => ! [C] : 
                  ( m1_subset_1(C,k2_zfmisc_1(c1_2__domain_1,A))
                 => ( ( k1_mcart_1(B) = k1_mcart_1(C)
                      & k2_mcart_1(B) = k2_mcart_1(C) )
                   => B = C ) ) ) ) )
   => ! [D] : 
        ( ~ v1_xboole_0(D)
       => ! [E] : 
            ( ~ v1_xboole_0(E)
           => ! [F] : 
                ( m1_subset_1(F,k2_zfmisc_1(D,E))
               => ! [G] : 
                    ( m1_subset_1(G,k2_zfmisc_1(D,E))
                   => ( ( k1_mcart_1(F) = k1_mcart_1(G)
                        & k2_mcart_1(F) = k2_mcart_1(G) )
                     => F = G ) ) ) ) ) ),
    introduced(definition,[new_symbol(c1_2__domain_1),file(domain_1,c1_2__domain_1)]),
    [interesting(0.8),axiom,file(domain_1,c1_2__domain_1)]).

fof(dh_c2_2__domain_1,definition,
    ( ( ~ v1_xboole_0(c2_2__domain_1)
     => ! [A] : 
          ( m1_subset_1(A,k2_zfmisc_1(c1_2__domain_1,c2_2__domain_1))
         => ! [B] : 
              ( m1_subset_1(B,k2_zfmisc_1(c1_2__domain_1,c2_2__domain_1))
             => ( ( k1_mcart_1(A) = k1_mcart_1(B)
                  & k2_mcart_1(A) = k2_mcart_1(B) )
               => A = B ) ) ) )
   => ! [C] : 
        ( ~ v1_xboole_0(C)
       => ! [D] : 
            ( m1_subset_1(D,k2_zfmisc_1(c1_2__domain_1,C))
           => ! [E] : 
                ( m1_subset_1(E,k2_zfmisc_1(c1_2__domain_1,C))
               => ( ( k1_mcart_1(D) = k1_mcart_1(E)
                    & k2_mcart_1(D) = k2_mcart_1(E) )
                 => D = E ) ) ) ) ),
    introduced(definition,[new_symbol(c2_2__domain_1),file(domain_1,c2_2__domain_1)]),
    [interesting(0.8),axiom,file(domain_1,c2_2__domain_1)]).

fof(dh_c3_2__domain_1,definition,
    ( ( m1_subset_1(c3_2__domain_1,k2_zfmisc_1(c1_2__domain_1,c2_2__domain_1))
     => ! [A] : 
          ( m1_subset_1(A,k2_zfmisc_1(c1_2__domain_1,c2_2__domain_1))
         => ( ( k1_mcart_1(c3_2__domain_1) = k1_mcart_1(A)
              & k2_mcart_1(c3_2__domain_1) = k2_mcart_1(A) )
           => c3_2__domain_1 = A ) ) )
   => ! [B] : 
        ( m1_subset_1(B,k2_zfmisc_1(c1_2__domain_1,c2_2__domain_1))
       => ! [C] : 
            ( m1_subset_1(C,k2_zfmisc_1(c1_2__domain_1,c2_2__domain_1))
           => ( ( k1_mcart_1(B) = k1_mcart_1(C)
                & k2_mcart_1(B) = k2_mcart_1(C) )
             => B = C ) ) ) ),
    introduced(definition,[new_symbol(c3_2__domain_1),file(domain_1,c3_2__domain_1)]),
    [interesting(0.8),axiom,file(domain_1,c3_2__domain_1)]).

fof(dh_c4_2__domain_1,definition,
    ( ( m1_subset_1(c4_2__domain_1,k2_zfmisc_1(c1_2__domain_1,c2_2__domain_1))
     => ( ( k1_mcart_1(c3_2__domain_1) = k1_mcart_1(c4_2__domain_1)
          & k2_mcart_1(c3_2__domain_1) = k2_mcart_1(c4_2__domain_1) )
       => c3_2__domain_1 = c4_2__domain_1 ) )
   => ! [A] : 
        ( m1_subset_1(A,k2_zfmisc_1(c1_2__domain_1,c2_2__domain_1))
       => ( ( k1_mcart_1(c3_2__domain_1) = k1_mcart_1(A)
            & k2_mcart_1(c3_2__domain_1) = k2_mcart_1(A) )
         => c3_2__domain_1 = A ) ) ),
    introduced(definition,[new_symbol(c4_2__domain_1),file(domain_1,c4_2__domain_1)]),
    [interesting(0.8),axiom,file(domain_1,c4_2__domain_1)]).

fof(antisymmetry_r2_hidden,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    [interesting(0.9),axiom,file(hidden,r2_hidden)]).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]).

fof(fc1_xboole_0,theorem,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,fc1_xboole_0)]).

fof(t1_subset,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    [interesting(0.9),axiom,file(subset,t1_subset)]).

fof(commutativity_k2_tarski,theorem,(
    ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ),
    file(tarski,k2_tarski),
    [interesting(0.9),axiom,file(tarski,k2_tarski)]).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_1)]).

fof(dt_k1_tarski,axiom,(
    $true ),
    file(tarski,k1_tarski),
    [interesting(0.9),axiom,file(tarski,k1_tarski)]).

fof(dt_k2_tarski,axiom,(
    $true ),
    file(tarski,k2_tarski),
    [interesting(0.9),axiom,file(tarski,k2_tarski)]).

fof(dt_k2_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k2_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_1)]).

fof(dt_c1_2__domain_1,assumption,(
    ~ v1_xboole_0(c1_2__domain_1) ),
    introduced(assumption,[file(domain_1,c1_2__domain_1)]),
    [interesting(0.8),axiom,file(domain_1,c1_2__domain_1)]).

fof(dt_c2_2__domain_1,assumption,(
    ~ v1_xboole_0(c2_2__domain_1) ),
    introduced(assumption,[file(domain_1,c2_2__domain_1)]),
    [interesting(0.8),axiom,file(domain_1,c2_2__domain_1)]).

fof(fc2_subset_1,theorem,(
    ! [A] : ~ v1_xboole_0(k1_tarski(A)) ),
    file(subset_1,fc2_subset_1),
    [interesting(0.9),axiom,file(subset_1,fc2_subset_1)]).

fof(fc3_subset_1,theorem,(
    ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ),
    file(subset_1,fc3_subset_1),
    [interesting(0.9),axiom,file(subset_1,fc3_subset_1)]).

fof(fc4_subset_1,theorem,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ),
    file(subset_1,fc4_subset_1),
    [interesting(0.9),axiom,file(subset_1,fc4_subset_1)]).

fof(rc1_xboole_0,theorem,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,rc1_xboole_0)]).

fof(rc2_xboole_0,theorem,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,rc2_xboole_0)]).

fof(t2_subset,theorem,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    [interesting(0.9),axiom,file(subset,t2_subset)]).

fof(t6_boole,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    [interesting(0.9),axiom,file(boole,t6_boole)]).

fof(t7_boole,theorem,(
    ! [A,B] : 
      ~ ( r2_hidden(A,B)
        & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    [interesting(0.9),axiom,file(boole,t7_boole)]).

fof(t8_boole,theorem,(
    ! [A,B] : 
      ~ ( v1_xboole_0(A)
        & A != B
        & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    [interesting(0.9),axiom,file(boole,t8_boole)]).

fof(dt_k1_mcart_1,axiom,(
    $true ),
    file(mcart_1,k1_mcart_1),
    [interesting(0.9),axiom,file(mcart_1,k1_mcart_1)]).

fof(dt_k2_mcart_1,axiom,(
    $true ),
    file(mcart_1,k2_mcart_1),
    [interesting(0.9),axiom,file(mcart_1,k2_mcart_1)]).

fof(dt_k4_tarski,axiom,(
    $true ),
    file(tarski,k4_tarski),
    [interesting(0.9),axiom,file(tarski,k4_tarski)]).

fof(dt_c3_2__domain_1,assumption,(
    m1_subset_1(c3_2__domain_1,k2_zfmisc_1(c1_2__domain_1,c2_2__domain_1)) ),
    introduced(assumption,[file(domain_1,c3_2__domain_1)]),
    [interesting(0.8),axiom,file(domain_1,c3_2__domain_1)]).

fof(dt_c4_2__domain_1,assumption,(
    m1_subset_1(c4_2__domain_1,k2_zfmisc_1(c1_2__domain_1,c2_2__domain_1)) ),
    introduced(assumption,[file(domain_1,c4_2__domain_1)]),
    [interesting(0.8),axiom,file(domain_1,c4_2__domain_1)]).

fof(fc1_zfmisc_1,theorem,(
    ! [A,B] : ~ v1_xboole_0(k4_tarski(A,B)) ),
    file(zfmisc_1,fc1_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,fc1_zfmisc_1)]).

fof(d5_tarski,definition,(
    ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ),
    file(tarski,d5_tarski),
    [interesting(0.9),axiom,file(tarski,d5_tarski)]).

fof(t23_mcart_1,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(A,k2_zfmisc_1(B,C))
     => A = k4_tarski(k1_mcart_1(A),k2_mcart_1(A)) ) ),
    file(mcart_1,t23_mcart_1),
    [interesting(0.9),axiom,file(mcart_1,t23_mcart_1)]).

fof(e1_2__domain_1,plain,
    ( k4_tarski(k1_mcart_1(c3_2__domain_1),k2_mcart_1(c3_2__domain_1)) = c3_2__domain_1
    & k4_tarski(k1_mcart_1(c4_2__domain_1),k2_mcart_1(c4_2__domain_1)) = c4_2__domain_1 ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__domain_1,dt_c2_2__domain_1,dt_c3_2__domain_1,dt_c4_2__domain_1])],[dt_k1_xboole_0,fc1_xboole_0,commutativity_k2_tarski,existence_m1_subset_1,dt_k1_tarski,dt_k2_tarski,dt_m1_subset_1,dt_c1_2__domain_1,dt_c2_2__domain_1,fc2_subset_1,fc3_subset_1,fc4_subset_1,rc1_xboole_0,rc2_xboole_0,t2_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,dt_k1_mcart_1,dt_k2_mcart_1,dt_k2_zfmisc_1,dt_k4_tarski,dt_c3_2__domain_1,dt_c4_2__domain_1,fc1_zfmisc_1,t1_subset,t7_boole,d5_tarski,t23_mcart_1]),
    [interesting(0.8),file(domain_1,e1_2__domain_1),[file(domain_1,e1_2__domain_1)]]).

fof(e2_2__domain_1,plain,
    ( ( k1_mcart_1(c3_2__domain_1) = k1_mcart_1(c4_2__domain_1)
      & k2_mcart_1(c3_2__domain_1) = k2_mcart_1(c4_2__domain_1) )
   => c3_2__domain_1 = c4_2__domain_1 ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__domain_1,dt_c2_2__domain_1,dt_c3_2__domain_1,dt_c4_2__domain_1])],[antisymmetry_r2_hidden,dt_k1_xboole_0,fc1_xboole_0,t1_subset,commutativity_k2_tarski,existence_m1_subset_1,dt_k1_tarski,dt_k2_tarski,dt_k2_zfmisc_1,dt_m1_subset_1,dt_c1_2__domain_1,dt_c2_2__domain_1,fc2_subset_1,fc3_subset_1,fc4_subset_1,rc1_xboole_0,rc2_xboole_0,t2_subset,t6_boole,t7_boole,t8_boole,dt_k1_mcart_1,dt_k2_mcart_1,dt_k4_tarski,dt_c3_2__domain_1,dt_c4_2__domain_1,fc1_zfmisc_1,d5_tarski,e1_2__domain_1]),
    [interesting(0.8),file(domain_1,e2_2__domain_1),[file(domain_1,e2_2__domain_1)]]).

fof(i4_2__domain_1,theorem,(
    $true ),
    introduced(tautology,[file(domain_1,i4_2__domain_1)]),
    [interesting(0.8),trivial,file(domain_1,i4_2__domain_1)]).

fof(i3_2__domain_1,plain,
    ( ( k1_mcart_1(c3_2__domain_1) = k1_mcart_1(c4_2__domain_1)
      & k2_mcart_1(c3_2__domain_1) = k2_mcart_1(c4_2__domain_1) )
   => c3_2__domain_1 = c4_2__domain_1 ),
    inference(conclusion,[status(thm),assumptions([dt_c1_2__domain_1,dt_c2_2__domain_1,dt_c3_2__domain_1,dt_c4_2__domain_1])],[e2_2__domain_1,i4_2__domain_1]),
    [interesting(0.8),file(domain_1,i3_2__domain_1),[file(domain_1,i3_2__domain_1)]]).

fof(i3_2_tmp__domain_1,plain,
    ( ( m1_subset_1(c3_2__domain_1,k2_zfmisc_1(c1_2__domain_1,c2_2__domain_1))
      & m1_subset_1(c4_2__domain_1,k2_zfmisc_1(c1_2__domain_1,c2_2__domain_1)) )
   => ( ( k1_mcart_1(c3_2__domain_1) = k1_mcart_1(c4_2__domain_1)
        & k2_mcart_1(c3_2__domain_1) = k2_mcart_1(c4_2__domain_1) )
     => c3_2__domain_1 = c4_2__domain_1 ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_2__domain_1,dt_c2_2__domain_1]),discharge_asm(discharge,[dt_c3_2__domain_1,dt_c4_2__domain_1])],[dt_c3_2__domain_1,dt_c4_2__domain_1,i3_2__domain_1]),
    [interesting(0.8),i2_2__domain_1]).

fof(i2_2__domain_1,plain,(
    ! [A] : 
      ( m1_subset_1(A,k2_zfmisc_1(c1_2__domain_1,c2_2__domain_1))
     => ! [B] : 
          ( m1_subset_1(B,k2_zfmisc_1(c1_2__domain_1,c2_2__domain_1))
         => ( ( k1_mcart_1(A) = k1_mcart_1(B)
              & k2_mcart_1(A) = k2_mcart_1(B) )
           => A = B ) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_2__domain_1,dt_c2_2__domain_1])],[i3_2_tmp__domain_1,dh_c3_2__domain_1,dh_c4_2__domain_1]),
    [interesting(0.8),file(domain_1,i2_2__domain_1),[file(domain_1,i2_2__domain_1)]]).

fof(i2_2_tmp__domain_1,plain,
    ( ~ v1_xboole_0(c2_2__domain_1)
   => ! [A] : 
        ( m1_subset_1(A,k2_zfmisc_1(c1_2__domain_1,c2_2__domain_1))
       => ! [B] : 
            ( m1_subset_1(B,k2_zfmisc_1(c1_2__domain_1,c2_2__domain_1))
           => ( ( k1_mcart_1(A) = k1_mcart_1(B)
                & k2_mcart_1(A) = k2_mcart_1(B) )
             => A = B ) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_2__domain_1]),discharge_asm(discharge,[dt_c2_2__domain_1])],[dt_c2_2__domain_1,i2_2__domain_1]),
    [interesting(0.8),i1_2__domain_1]).

fof(i1_2__domain_1,plain,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_subset_1(B,k2_zfmisc_1(c1_2__domain_1,A))
         => ! [C] : 
              ( m1_subset_1(C,k2_zfmisc_1(c1_2__domain_1,A))
             => ( ( k1_mcart_1(B) = k1_mcart_1(C)
                  & k2_mcart_1(B) = k2_mcart_1(C) )
               => B = C ) ) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_2__domain_1])],[i2_2_tmp__domain_1,dh_c2_2__domain_1]),
    [interesting(0.8),file(domain_1,i1_2__domain_1),[file(domain_1,i1_2__domain_1)]]).

fof(i1_2_tmp__domain_1,plain,
    ( ~ v1_xboole_0(c1_2__domain_1)
   => ! [A] : 
        ( ~ v1_xboole_0(A)
       => ! [B] : 
            ( m1_subset_1(B,k2_zfmisc_1(c1_2__domain_1,A))
           => ! [C] : 
                ( m1_subset_1(C,k2_zfmisc_1(c1_2__domain_1,A))
               => ( ( k1_mcart_1(B) = k1_mcart_1(C)
                    & k2_mcart_1(B) = k2_mcart_1(C) )
                 => B = C ) ) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_2__domain_1])],[dt_c1_2__domain_1,i1_2__domain_1]),
    [interesting(1),t12_domain_1]).

fof(t12_domain_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( m1_subset_1(C,k2_zfmisc_1(A,B))
             => ! [D] : 
                  ( m1_subset_1(D,k2_zfmisc_1(A,B))
                 => ( ( k1_mcart_1(C) = k1_mcart_1(D)
                      & k2_mcart_1(C) = k2_mcart_1(D) )
                   => C = D ) ) ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_2_tmp__domain_1,dh_c1_2__domain_1]),
    [interesting(1),file(domain_1,t12_domain_1),[file(domain_1,t12_domain_1)]]).